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We consider a non-durable good monopolist that collects data on its customers in order to profile them and subsequently practice price discrimination on returning customers. The monopolist's price discrimination scheme is leaky, in the sense that an endogenous fraction of consumers may choose to incur a privacy cost to conceal their identity in future purchases. We characterize the Markov Perfect Equilibrium of the dynamic game under two alternative customer profiling regimes: full information acquisition (FIA) and Purchase History Information (PHI). In both cases, we find that, contrary to what could have been expected, the aggregate profit is not monotonically increasing in the level of the privacy cost but a U-shaped function of it, leading to ambiguous profit effects: A reduction in privacy costs increase the fraction of customers who choose to be anonymous (detrimental profit effect) but it also softens the firm's introductory price, reducing the pace at which prices targeted to new customers fall over time (positive profit effect). When comparing results under FIA and PHI, we find that market expansion is faster and more customers conceal their identity under FIA than under PHI. Equilibrium profits are also higher in the FIA case. Although equilibrium profits are U-shaped functions of the privacy cost in both profiling regimes, they tend to be globally decreasing under PHI, while being globally increasing with the privacy cost under FIA.